By Silvia Biasotti, Bianca Falcidieno, Daniela Giorgi, Michela Spagnuolo
This ebook is a advisor for researchers and practitioners to the hot frontiers of 3D form research and the complicated mathematical instruments so much tools depend on. the objective reader comprises scholars, researchers and execs with an undergraduate arithmetic history, who desire to comprehend the math at the back of form research. The authors commence with a short evaluation of simple options in geometry, topology, differential geometry, and continue to complicated notions of algebraic topology, constantly maintaining a tally of the appliance of the idea, via examples of form research tools corresponding to 3D segmentation, correspondence, and retrieval. a couple of learn recommendations within the box come from advances in natural and utilized arithmetic, in addition to from the re-reading of classical theories and their version to the discrete environment. In a global the place disciplines (fortunately) have blurred obstacles, the authors think that this consultant might help to bridge the space among idea and perform.
Read or Download Mathematical Tools for Shape Analysis and Description (Synthesis Lectures on Computer Graphics and Animation) PDF
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Additional resources for Mathematical Tools for Shape Analysis and Description (Synthesis Lectures on Computer Graphics and Animation)
Pose transfer e pose transfer approach proposed in Lévy  is based on the Fourier decomposition of the manifold embedding coordinates. ere the “layout” (pose) of the shape X is transferred to the shape Y while preserving the geometric details of Y . In a formal way, given two shapes X and Y embedded in R3 with the corresponding harmonic bases iX and iY , and the corP P responding Fourier decompositions of the embeddings X D i 1 ai iX and Y D i 1 bi iY , P P Y Y a new shape Z is composed according to the rule Z D N iD1 ai i C i>N bi i with the ﬁrst N low frequency coeﬃcients taken from X , and higher frequencies taken from Y .
In the Euclidean plane R2 and the Euclidean space R3 , an example of isometries are geometric congruences, which include rigid motions (translations, rotations) and reﬂections. If dX is the intrinsic metric of X , isometries preserve geodesic distances. 1, the hand bending is the result of an isometric transformation , and the geodesic distance between points p and q on the hand on the left is the same as the geodesic distance of their images p0 and q0 on the hand on the right, after bending. 1: Isometric transformation of the space X into Y .
3, the picture on the right side shows that there are two shortest paths between P and Q as the distances to travel around the Gaussian bell are the same. 3: Left: the blue line represents the geodesic segment between the points P and Q while the yellow one is a possible path whose length is not minimal. Right: the shortest path is not necessarily unique. Geodesic distance With these premises, the geodesic distance between two points is deﬁned as the length of a shortest path between the two points.